Theorem isoml 39869

Description: The predicate "is an orthomodular lattice." (Contributed by NM, 18-Sep-2011.)

Hypotheses

Ref	Expression
isoml.b	⊢ 𝐵 = (Base‘𝐾)
isoml.l	⊢ ≤ = (le‘𝐾)
isoml.j	⊢ ∨ = (join‘𝐾)
isoml.m	⊢ ∧ = (meet‘𝐾)
isoml.o	⊢ ⊥ = (oc‘𝐾)

Assertion

Ref	Expression
isoml	⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))

Distinct variable groups:   𝑥,𝑦,𝐵   𝑥,𝐾,𝑦

Allowed substitution hints:   ∨ (𝑥,𝑦)   ≤ (𝑥,𝑦)   ∧ (𝑥,𝑦)   ⊥ (𝑥,𝑦)

Proof of Theorem isoml

Dummy variable 𝑘 is distinct from all other variables.

Step	Hyp	Ref	Expression
1		fveq2 6871	. . . 4 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = (Base‘𝐾))
2		isoml.b	. . . 4 ⊢ 𝐵 = (Base‘𝐾)
3	1, 2	eqtr4di 2818	. . 3 ⊢ (𝑘 = 𝐾 → (Base‘𝑘) = 𝐵)
4		fveq2 6871	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = (le‘𝐾))
5		isoml.l	. . . . . . 7 ⊢ ≤ = (le‘𝐾)
6	4, 5	eqtr4di 2818	. . . . . 6 ⊢ (𝑘 = 𝐾 → (le‘𝑘) = ≤ )
7	6	breqd 5115	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑥(le‘𝑘)𝑦 ↔ 𝑥 ≤ 𝑦))
8		fveq2 6871	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = (join‘𝐾))
9		isoml.j	. . . . . . . 8 ⊢ ∨ = (join‘𝐾)
10	8, 9	eqtr4di 2818	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (join‘𝑘) = ∨ )
11		eqidd 2766	. . . . . . 7 ⊢ (𝑘 = 𝐾 → 𝑥 = 𝑥)
12		fveq2 6871	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = (meet‘𝐾))
13		isoml.m	. . . . . . . . 9 ⊢ ∧ = (meet‘𝐾)
14	12, 13	eqtr4di 2818	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → (meet‘𝑘) = ∧ )
15		eqidd 2766	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → 𝑦 = 𝑦)
16		fveq2 6871	. . . . . . . . . 10 ⊢ (𝑘 = 𝐾 → (oc‘𝑘) = (oc‘𝐾))
17		isoml.o	. . . . . . . . . 10 ⊢ ⊥ = (oc‘𝐾)
18	16, 17	eqtr4di 2818	. . . . . . . . 9 ⊢ (𝑘 = 𝐾 → (oc‘𝑘) = ⊥ )
19	18	fveq1d 6873	. . . . . . . 8 ⊢ (𝑘 = 𝐾 → ((oc‘𝑘)‘𝑥) = ( ⊥ ‘𝑥))
20	14, 15, 19	oveq123d 7421	. . . . . . 7 ⊢ (𝑘 = 𝐾 → (𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)) = (𝑦 ∧ ( ⊥ ‘𝑥)))
21	10, 11, 20	oveq123d 7421	. . . . . 6 ⊢ (𝑘 = 𝐾 → (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))
22	21	eqeq2d 2776	. . . . 5 ⊢ (𝑘 = 𝐾 → (𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))) ↔ 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥)))))
23	7, 22	imbi12d 347	. . . 4 ⊢ (𝑘 = 𝐾 → ((𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
24	3, 23	raleqbidv 3339	. . 3 ⊢ (𝑘 = 𝐾 → (∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
25	3, 24	raleqbidv 3339	. 2 ⊢ (𝑘 = 𝐾 → (∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥)))) ↔ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))
26		df-oml 39810	. 2 ⊢ OML = {𝑘 ∈ OL ∣ ∀𝑥 ∈ (Base‘𝑘)∀𝑦 ∈ (Base‘𝑘)(𝑥(le‘𝑘)𝑦 → 𝑦 = (𝑥(join‘𝑘)(𝑦(meet‘𝑘)((oc‘𝑘)‘𝑥))))}
27	25, 26	elrab2 3657	1 ⊢ (𝐾 ∈ OML ↔ (𝐾 ∈ OL ∧ ∀𝑥 ∈ 𝐵 ∀𝑦 ∈ 𝐵 (𝑥 ≤ 𝑦 → 𝑦 = (𝑥 ∨ (𝑦 ∧ ( ⊥ ‘𝑥))))))

Syntax hints:   → wi 4   ↔ wb 209   ∧ wa 400   = wceq 1563   ∈ wcel 2145  ∀wral 3079   class class class wbr 5104  ‘cfv 6525  (class class class)co 7400  Basecbs 17257  lecple 17305  occoc 17306  joincjn 18355  meetcmee 18356  OLcol 39805  OMLcoml 39806

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1818  ax-4 1832  ax-5 1933  ax-6 1990  ax-7 2031  ax-8 2147  ax-9 2155  ax-ext 2737

This theorem depends on definitions:  df-bi 210  df-an 401  df-or 861  df-3an 1103  df-tru 1566  df-fal 1576  df-ex 1803  df-sb 2094  df-clab 2744  df-cleq 2757  df-clel 2840  df-ral 3080  df-rab 3418  df-v 3459  df-dif 3910  df-un 3912  df-ss 3924  df-nul 4289  df-if 4484  df-sn 4586  df-pr 4588  df-op 4592  df-uni 4868  df-br 5105  df-iota 6481  df-fv 6533  df-ov 7403  df-oml 39810

This theorem is referenced by:  isomliN  39870  omlol  39871  omllaw  39874